$\mathrm{\Gamma }$-convergence of quadratic functionals with non uniformly elliptic conductivity matrices

We investigate the homogenization through $\mathrm{\Gamma }$-convergence for the $L^2(\mathrm{\Omega })$-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix $A^{\ast }$ is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional $1$-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of $A^{\ast }$ is provided which is a generalization of the classical laminate formula. We construct a two-dimensional counter-example which shows an anomalous asymptotic behaviour of the conductivity functional.